Shuai
Yang
,
Fengrui
Sui
,
Yucheng
Liu
,
Ruijuan
Qi
*,
Xiaoyu
Feng
,
Shangwei
Dong
,
Pingxiong
Yang
and
Fangyu
Yue
*

Key Laboratory of Polar Materials and Devices (MOE), Department of Electronics, East China Normal University, Shanghai 200241, China. E-mail: fyyue@ee.ecnu.edu.cn; rjqi@ee.ecnu.edu.cn

Received
7th June 2023
, Accepted 24th July 2023

First published on 25th July 2023

Low-symmetric GeTe semiconductors have attracted wide-ranging attention due to their excellent optical and thermal properties, but only a few research studies are available on their in-plane optical anisotropic nature that is crucial for their applications in optoelectronic and thermoelectric devices. Here, we investigate the optical interactions of anisotropy in GeTe using polarization-resolved Raman spectroscopy and first-principles calculations. After determining both armchair and zigzag directions in GeTe crystals by transmission electron microscopy, we found that the Raman intensity of the two main vibrational modes had a strong in-plane anisotropic nature; the one at ∼88.1 cm^{−1} can be used to determine the crystal orientation, and the other at ∼124.6 cm^{−1} can reveal a series of temperature-dependent phase transitions. These results provide a general approach for the investigation of the anisotropy of light–matter interactions in low-symmetric layered materials, benefiting the design and application of optoelectronic, anisotropic thermoelectric, and phase-transition memory devices based on bulk GeTe.

In this work, we investigate the in-plane anisotropy of low-symmetric rhombohedral GeTe crystals using high-angle annular dark-field scanning transmission electron microscopy (HAADF-STEM), Raman spectroscopy, and first-principles calculations. By analyzing the vibrational modes via the polarization- and temperature-dependent Raman results under parallel and cross configurations, we observed up to four Raman active modes. Based on this, the crystal orientation could be determined and the symmetry of the crystal structure could be understood. Furthermore, a strong four-phonon coupling process was analyzed during the change in temperature and the temperature-induced phase transitions between the α, γ, and β phases could be determined in GeTe.

The electronic structure of bulk GeTe was calculated using the Kohn–Sham density functional theory (KS-DFT).^{31} The real space electronic structure calculator (RESCU) was used to calculate the phonon modes and the phonon scattering relationship for bulk GeTe.^{32,33} The exchange–correlation potential was described by the generalized gradient approximation (GGA) of the Perdew–Burke–Ernzerhof functional (PBE).^{34} The Monkhorst–Pack scheme with 7 × 7 × 7 mesh was used to sample k-points of the Brillouin zone of bulk GeTe. The physical quantities were expanded using the real space lattice method, and the self-consistent field procedure was performed until the global charge variation was less than 10^{−5}. Lattice-dynamical properties were obtained for the Γ-point using the direct-force constant approach.

We performed Raman measurements on the GeTe crystal at different temperatures. Fig. 2a shows the Raman spectra of bulk GeTe at 300 K and ∼80 K (more details in Fig. S3†) with a similar curve shape. A fit procedure on the curve of 300 K roughly presented five peaks, which were denoted as A, B, C, D, and E at 88.1 cm^{−1}, 124.6 cm^{−1}, 142.2 cm^{−1}, 158.4 cm^{−1}, and 225.5 cm^{−1}, respectively. The two characteristic peaks, A and B, originated from the Ge–Te vibration; the former among these is the double-degenerate E symmetry transverse/longitudinal optical modes, and the latter is the A_{1} symmetry transverse optical mode.^{36–39} Peak C is due to the long-range interactions between crystalline Te, and peak D is triggered by the vibrational density of states of the long Te chains in disorder.^{37,40} Peak E may be assigned to the antisymmetric stretching mode of the GeTe_{4} tetrahedra.^{41} The calculated Raman peaks by the density functional perturbation theory (DFPT)^{42} are also shown with black vertical lines in Fig. 2a. Fig. 2b depicts the calculated phonon dispersion results. There are three acoustic waves with zero frequency at the Γ point, and the rest are optical waves. The phonon spectrum has negligible tiny virtual frequencies near the Γ point, indicating good dynamical stability of the model structure. Based on the group theory, GeTe has a total of two phonon modes, E (96.3 cm^{−1}) and A_{1} (121.8 cm^{−1}), and both are Raman- and infrared-active. By comparing with the theoretical and experimental (at ∼80 K) values in Fig. 2a, where the values of E and A_{1} modes are experimentally 99 and 127 cm^{−1}, respectively, we can find a frequency difference in the two main modes, i.e., ∼2.7 and ∼5.2 cm^{−1} blue shifts of E and A_{1} modes, respectively. Inconsistency was also observed in low-symmetric ReSe_{2},^{43} whose reason was ascribed to the weak interlayer interaction due to the presence of both in-plane and out-of-plane components in their A_{g} vibrational modes. Therefore, the frequency shift in the A_{1} mode may be understood as a competition between the interlayer coupling and the dielectric shielding. However, the maximum frequency shift of only 4 cm^{−1} in ReSe_{2} was significantly different from that observed with the E mode. This suggests that other potential influences may be involved here, e.g., the effect of temperature.^{44} Notice that according to the DFT calculation (Fig. S1†), the direct E_{g} transition for bulk GeTe is located at the L point (E_{g} = 0.71 eV), and a slightly larger E_{g} can be seen at the T point (0.88 eV). The highest valence band (VB) and the lowest conduction band (CB) are dominated by the 4p orbital of Ge and the 5p orbital of Te, respectively. These observations are consistent with previous work^{28} and verify the reliability and accuracy of the calculated results.

Polarization-resolved Raman measurements were further performed. Fig. 3a–d show the polar plot of the Raman intensity for the two Raman modes, E (88.1 cm^{−1}) and A_{1} (124.6 cm^{−1}). The results for other modes are shown in Fig. S2;† the Raman modes exhibit different degrees of anisotropy. The E mode exhibits strong anisotropy in both cross (Fig. 3a) and parallel (Fig. 3c) configurations with a period of 90°, but with an angular difference of ∼45° for the intensity maximum and minimum values. The mode of A_{1} exhibits a two-lobe shape only in the cross configuration (Fig. 3b) with a period of 180°. While the behavior in the parallel configuration exhibits an asymmetric case (Fig. 3d), the poles correspond to the same angles as in the cross configuration. One of the reasons for this phenomenon may be the non-normal incident of the laser onto the surface of GeTe.

Fig. 3 (a)–(d) Polar plots of the peak intensity of E and A_{1} modes under the polarization configurations (parallel or cross). Stars – experimental values; green curves – fit results. |

To better understand the results of polarization-resolved Raman spectroscopy of GeTe, the second-order Raman tensor for the Raman mode is taken into account. According to the classical Placzek approximation,^{45} the Raman intensity can be written as,

I ∝ |e_{i}·R·e_{s}|^{2} | (1) |

In the current geometrical configuration, e_{i} and e_{s} are in the x–z plane. Thus, the incident polarization angle of θ is set to the crystal axis, e_{i} = (cosθ, 0, sinθ), while e_{s} = (cosθ, 0, sinθ)^{T} (parallel polarization configuration) and (−sinθ, 0, cosθ)^{T} (cross-polarization configuration). Table 2 shows the derived intensity for A_{1} and E modes in parallel and cross-polarization configurations obtained from eqn (1). The measured data for E (88.1 cm^{−1}) mode exhibited a tetrad shape, which could be traced well by the equation listed in the table; also see Fig. 3a and c. However, the modes of 124.6 cm^{−1} (and 142.2 cm^{−1}) are poorly fitted, which may be due to the light absorption and birefringence effects mentioned above. There was no obvious rule for the 158.4 cm^{−1} mode, probably due to the vibration of disordered long Te chains. The period of 225.5 cm^{−1} in both configurations is 90°, which is the same as that of 88.1 cm^{−1} corresponding to the antisymmetric stretching mode of the GeTe_{4} tetrahedron. Thus, for GeTe, the maxima of the E Raman intensity can be aligned along 0° in the parallel polarization configuration. This indicates that 0 or 90° corresponds to either the x- or y-axis of the GeTe crystal, indicating the crystal orientation.

Fig. 4a shows the Raman spectra of GeTe during the temperature range of 300–680 K (see Fig. S3† for results at low temperature). With the increase in temperature, the Raman peaks of the three strong modes showed different red-shifts and broadenings, which were generally related to anharmonic phonon interactions, electron–phonon coupling, or thermal expansion.^{46} For clarity, Fig. 4b–d depict the temperature dependence of the peak intensity, position, and full-width at half-maximum (FWHM) of the three Raman modes. From Fig. 4b, we can see that the 88.1 cm^{−1} mode disappears beyond ∼600 K, and the modes at 124.6 cm^{−1} and 142.2 cm^{−1} were not visible after 670 K; in addition, the clear inflection points (different slopes or disappearing) can be seen at 600 or 670 K (shadow zone), which may suggest that the GeTe undergoes different temperature-induced phase transitions. In Fig. 4c, all three modes were almost linearly red-shifted with an increase in temperature, which could be fitted with the following expression,

ω(T) = ω_{0} + χT | (2) |

Simultaneously, as shown in Fig. 4d, the FWHM of three modes shows different trends with increasing temperature, where the FWHM of 88.1 and 124.6 cm^{−1} modes shows a similar temperature-dependent trend but with a slight decrease in the temperature range of 300–450 K. It is well known that the smaller the FWHM is, the more regular the crystal vibration is and the more stable the crystal structure is. On the contrary, after 450 K the FWHM increases significantly, and the crystal structure starts to become unstable; additionally, the E and A_{1} modes start to present thermal-related broadening. The 142.2 cm^{−1} mode showed an approximately linear evolution with the temperature here, which is mainly due to the presence of the homopolar bonding of Te–Te, a trend that is prevalent in Te compounds, such as BiTe.^{51} When the temperature reaches 600 K, the E mode at 88.1 cm^{−1} disappears, and the A_{1} mode at 124.6 cm^{−1} maintains a slightly linear increase till the temperature is beyond 670 K, where the Raman pattern approaches a curve with a very low signal-to-noise ratio without the appearance of evident peaks. A comparison with the results of Raman intensity and shift shows that these behaviors are consistent with the three key parameters.

It has been reported that GeTe can spontaneously transit between the polar α-phase (rhombohedral, low-temperature phase) and the β-phase (cubic, high-temperature phase) at the Curie temperature of ∼625 K,^{52} and the phase transition temperature of GeTe is highly influenced by its stoichiometric ratio. For instance, Rinaldi et al. found that the tendency of Te to detach from GeTe at high temperatures leads to a change in its stoichiometric ratio.^{17} Therefore, we believe that the GeTe has started to undergo a phase change, when the temperature is close to ∼600 K, and may complete the phase transition process after ∼670 K by referring to the similar temperature-dependent way and the almost simultaneous disappearance of the modes of 124.6 and 142.2 cm^{−1}. Thus, by combining with the phase diagram involving the composition ratio-dependent phase transition dynamics of GeTe,^{10} we suggest that in the temperature range of 600–670 K, the status of GeTe comprises mixed α, γ, and β phases, i.e., first from α phase to γ phase and then to β phase with the increase in temperature. For the high-temperature stable β phase with a sodium chloride structure, the previous report gave a phase transition temperature of 670 K,^{53} and a close phase transition temperature of 673 K was also reported.^{10}

Notice that the FWHM broadening of the Raman peaks can be fitted with the higher-order phonon scattering model proposed by Balkanski et al.:^{54}

(3) |

Raman shift (cm^{−1}) |
A (cm^{−1}) |
B (cm^{−1}) |
C (cm^{−1}) |
---|---|---|---|

88 (E mode) | 9.78 | 3.64 × 10^{−12} |
−3.27 × 10^{−25} |

124 (A_{1} mode) |
8.25 | 3.99 × 10^{−13} |
1.87 × 10^{−25} |

142 (Te–Te) | 5.72 | 3.91 × 10^{−12} |
5.06 × 10^{−25} |

Due to the lack of well-fitted data using the higher-order phonon scattering model, for phase-change materials, the temperature dependence of the FWHM of Raman peaks can be investigated by the model proposed by F. Jebari et al.,^{55}

(4) |

Parameters | 88 (cm^{−1}) |
124 (cm^{−1}) |
142 (cm^{−1}) |
---|---|---|---|

A′ | 6.41 | 5.14 | 3.79 |

B′ | 4.29 × 10^{−2} |
3.01 × 10^{−2} |
3.22 × 10^{−2} |

C′ | −44.45 | −30.36 | −22.81 |

E
_{a} (kJ mol^{−1}) |
8.19 | 7.46 | 7.75 |

Based on the FWHM of Raman peaks at different temperatures, we can semi-quantitatively estimate the thermal conductivity (K) of GeTe.^{58}

(5) |

Where C_{v}, , and τ_{T} are heat capacity under constant volume, averaged acoustic phonon velocity, and the thermal transport relaxation time, respectively. Under the assumed conditions, where C_{v} is 2k_{B} and is 5 km s^{−1},^{4,59} the thermal transport relaxation time can be approximated as the quasi-particle lifetime of the Raman optical phonon τ, . By referring to the FWHM (Γ) value in the range of 8–16 cm^{−1}, τ ranges from 0.33–0.66 ps. The range of the K value from eqn (5) can then be 5.5–11.0 W m^{−1} K^{−1}, of which the mean value is ∼8.25 W m^{−1} K^{−1}, which is extremely close to the previously-reported experimental value of 8.3 W m^{−1} K^{−1}.^{60}

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## Footnote |

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d3nr02678g |

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